Moving hook stretches edges in cloth sim
This is likely related to https://blender.stackexchange.com/a/83096/29586 - ie, the sequence of the modifiers in the modifier stack. Ensure the Hook modifiers are above the cloth sim modifier - otherwise the cloth will apply first (to the unmoved vertices) and then the hook modifiers will move those vertices after the sim - rather than the sim taking account of the new location of those vertices.



EDIT I've put together a test and this is working as I would expect - ie, with the hook modifiers 'above' the cloth simulation. Here's the setup :



The 'cloth' consists of a subdivided plane. I have added a Vertex Group named 'pin' and assigned two vertices on the top edge to it. I selected each of these same vertices in turn and used Ctrl+H to 'Hook to New Object' to create two new empties hooked to those vertices. Note : It's important to use the exact same vertices for the vertex group as is assigned to the hook.

Using keyframes I animated the right-hand empty.



This produced the following animation :



This only works with the hooks above the cloth simulation in the modifier stack.



Blend file included